This problem asked how many different ways there are to arrange tiles of 1 inch by 2 inches in an area of 20 inches by 2 inch.
For this problem, we started smaller to see if we could find a pattern and go from there. We started with a 2 inch wide by 4 inch long path since the problem gave the example of a 2 inch by 3 inch path. We set up a x and y table and kept making the path longer and seeing how many ways there were to rearrange it. As we looked back at the table, we realized that the numbers started looking like a very familiar pattern, the fibonacci sequence. We then realized that the length of the path would be the number of term in the sequence. We then predicted the next number and tested it out.
For our solution we got 10,946. We got this by finding the 20th term in the sequence.
At first I thought this problem was going to be very easy and that I could just find the number of times I could rearrange them in a 4 inch long path then multiply it by 5 since you could do that 20 times. Then once I started doing the x and y table it wasn’t that simple. Drawing out the different arrangements for the different lengths was also hard because I was scared that I missed one and I actually repeated 2 one time which kind of threw me off at first.
I think I did good in this problem even though it took me a little longer to find the answer. I kept trying different ways that I thought would work and wouldn’t in the end but I didn’t give up. I persevered through until I found the answer. I also feel like I was pretty creative in the way I found the answer, I kept looking for different ways since my initial ideas didn’t work. In the end, I came across the answer kind of by accident but I used my prior knowledge to make the x and y table and saw the pattern in that way.